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On Euclidean Ideal Classes.

Graves, Hester K.

Graves, Hester K.

2009

Abstract: In 1979, H.K.Lenstra generalized the idea of Euclidean algorithms to
Euclidean ideal classes. If a domain has a Euclidean algorithm, then it
is a principal ideal domain and has a trivial class group; if a Dedekind
domain has a Euclidean ideal class, then it has a cyclic class group gen-
erated by the Euclidean ideal class. Lenstra showed that if one assumes
the generalized Riemann hypothesis and a number field has a ring of in-
tegers with infinitely many units, then said ring has cyclic class group if
and only if it has a Euclidean ideal class.
Malcolm Harper’s dissertation built up general machinery that allows
one to show a given ring of integers (with infinitely many units) of a
number field with trivial class group is a Euclidean ring. In order to
build the machinery, Harper used the Large Sieve and the Gupta-Murty
bound.
This dissertation generalizes Harper’s work to the Euclidean ideal class
setting. In it, there is general machinery that allows one to show that a
number field with cyclic class group and a ring of integers with infinitely
many units has a Euclidean ideal class. In order to build this machinery, the Large Sieve and the Gupta-Murty bound needed to be generalized to
the ideal class situation. The first required class field theory; the second
required several asymptotic results on the sizes of sets of k-tuples.